Math 74 Midterm 2: Solutions
November 12, 2008
1. Let X and Y be sets.
(a) Show that P (X Y ) = P (X ) P (Y ).
(b) If |X | = n, |Y | = m, and |X Y | = , calculate |P (X ) P (Y )|.
(c) With the same nu
NOTES ON POINT-SET TOPOLOGY These notes introduce some basic concepts in the eld known as topology. Here we will only be concerned with point-set topology, essentially the topology of the set of real
MATH 74: HOMEWORK 9 COMMENTS Book Problem # 13 The least common multiple of two nonzero integers a and b is the unique positive integer m such that (i) m is a common multiple, i.e., a|m and b|m, (ii)
MATH 74 MIDTERM 2
All theorem references refer to the numbers on the note sheet. 1. A = cfw_1 and B = cfw_2 will work. [More generally, any two sets with the property that neither is a subset of the o
MATH 74 MIDTERM 2
1. Short reponse: 20 points For any set S , let P (S ) denote the set of all subsets of S . 1. [4 pts] Give an example of sets A and B such that P (A B ) = P (A) P (B ). You do not n
NOTES ON FUNCTIONS These notes will cover some terminology regarding functions not included in Solows book. You should read Appendix A.2 in the book before reading these notes. Denition 1. We say that
NOTES ON GROUP THEORY The goal of these notes is to introduce some of the basic concepts which come up in Math 113. As with the analysis/topology material, the emphasis will be placed on studying exam
NOTES ON LIMITS In these notes we give the denition of the limit of a function ; the material on sequences is in Appendix D.2 of the book. Our goal is to see how to use this denition to prove some of
NOTES ON REAL NUMBERS In these notes we will construct the set of real numbers. Why would we want to do this you may ask? Well, mathematicians want mathematics to be based on a solid foundation, such
NOTES ON RELATIONS These notes introduce the notion of a relation and, more importantly, the notion of an equivalence relation. Denition 1. A relation R between two sets A and B is a subset of the Car
NOTES ON SET THEORY The purpose of these notes is to cover some set theory terminology not included in Solows book. You should read Appendix A.1 in the book before reading these notes. The symbol := m
MATH 74 MIDTERM 2 NOTE SHEET
Theorem 1. A function f : X Y is invertible if and only if it is a bijection. Let Nk = cfw_x N : 1 x k . Theorem 2. If X is a nite set, then #(X ) = k if and only if there
MATH 74 MIDTERM 1
Short response [20 points] 1. Answer true or false. [2 points each] (a) If is a binary operation on a set S , and is not commutative, then for any a and b in S , it must be that a b
MATH 74 HOMEWORK 9 (DUE WEDNESDAY NOVEMBER 7)
Exercises 1,2. Let A = cfw_1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 1. Let X denote the set of subsets of A that have an odd number of elements. Let Y denote the se
MATH 74 HOMEWORK 10 (DUE WEDNESDAY NOVEMBER 28)
1(a). We rst prove that if 5 | a, then 5 | a2 . Suppose 5 | a. This means that there is q Z with a = 5q . Squaring both sides we see a2 = 25q 2 = 5(q 2
MATH 74 HOMEWORK 10 (DUE WEDNESDAY NOVEMBER 28)
1(a). Problem 15.2 on page 198 of Eccles. 1(b). Give an example of an integer n with the property that the statement of 1(a), with n substituted for 5,
MATH 74 HOMEWORK 11
1. Since gcd(a1 , b1 ) = 1, a theorem from class gives us integers M and N with M a1 + N b1 = 1. Multiplying through by b2 we obtain M a1 b2 + N b1 b2 = b2 , and using the fact tha
MATH 74 HOMEWORK 11 (DUE FRIDAY DECEMBER 7)
Note the nonstandard due date. 1. Suppose that a1 , a2 , b1 , b2 are positive integers, that (1) that (2) and that (3) a1 b2 = a2 b1 . Prove that a1 = a2 an
MATH 74 FINAL EXAM THEOREM SHEET You may take for granted that + and are binary operations on Z satisfying the commutative, associative, and distributive laws. You thus do not need to pay attention to
FINAL EXAM COMMENTS AND SOLUTIONS
1. Comments 1. This was problem 5 on Homework 3. 2. This is a variation on problem 5 of Homework 3. (We have seen operations with this property before; see e.g. probl
MATH 74 FINAL EXAM [SOME TYPOS CORRECTED]
Short response. [6 pts each] 1. Give an example of an associative binary operation on R that is not commutative. 2. Give an example of a commutative binary op
MATH 74 MIDTERM 1 SHEET
+ and are binary operations on R, and (A1) x + (y + z ) = (x + y ) + z for all x, y , and z in R (A2) x + y = y + x for all x and y in R (A3) 0 + x = x for all x in R (M1) (x y
MATH 74 MIDTERM 1
1(a). False. (No matter what is, and no matter what a S is, the equation a a = a a is always true, for example.) 1(b). True. For any integer k , the real number sin( k ) is an intege
Math 74 Final Exam: Solutions
December 16, 2008
1. (a) Let (X, d) be a metric space, and let (xn ) be a sequence in (X, d).
Dene what it means for (xn ) to be convergent.
(b) Use quantier negation to
Math 74 Final Exam
December 16th, 2008
Name
SID
Question
1
2
3
4
5
6
7
8
Score
1
Possible
10
11
6
6
10
6
5
7
61
1. (a) Let (X, d) be a metric space, and let (xn ) be a sequence in (X, d).
Dene what it
05/09/2003 FRI 15:28 FAX 6434330 MOFFITT LIBRARY 001
J. Horowrl'z.
MATHEMATICS 74: FINAL EXAM
December 18, 2002
Follow all of the instructions carefully. Make sure to give reasons if they are asde for
Section 6.6
Partial Orderings
Definition: Let R be a relation on A. Then R is a partial
order iff R is
reflexive
antisymmetric
and
transitive
(A, R) is called a partially ordered set or a poset.
_
Section 2: Reflexivity, Symmetry, and Transitivity
10.2.1
Definition: Let R be a binary relation on A. R is reflexive if for all x A, (x,x) R. (Equivalently, for all x e A, x R x.) R is symmetric if
The Well-Ordering Principle
The well-ordering principle is a concept which is equivalent to mathematical induction. In your textbook, there is a proof for how the well-ordering principle implies
the v
ax b mod m
Linear Congruences
Theorem 1. If (a, m) = 1, then the congruence ax b mod m phas exactly one solution
modulo m.
Constructive.
Solve the linear system
sa + tm = 1.
Then
sba + tbm = b.
So
sba